: What are the pros and cons of each technique? Do all
: those staples damage the integrity of the wood? Do you use
: stainless staples (do they exist)?

Both methods work fine. A one piece hull, sewn down the middle, takes much longer and is far more tedious than skinning with two pieces of fabric. OTOH, a two piece skinning job needs a rubrail to cover the seam, and that is another step you have to deal with. Also, it's not really historically accurate to have rubrails on a Greenland style kayak, if that matters to you at all. I have not noticed any appreciable weakening of the gunwales with the addition of the usual number of staples or screws.
Stainless steel staples are commonly available at Home Depot, Lowes and plenty of hardware stores. The boxes don't look much different than standard staples, so you have to look at them closely. Monel staples are better, but I haven't needed that extra durability, yet.